[Paper Review] Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming

Nai-Hui Chia, Tongyang Li+2 more|arXiv (Cornell University)|Jan 10, 2019

Complexity and Algorithms in Graphs28 references4 citations

TL;DR

This paper presents a classical, quantum-inspired sublinear-time algorithm for solving low-rank semidefinite programs (SDPs) using sampling-based data structures. By combining the matrix multiplicative weight framework with novel sampling techniques—weighted sampling for matrix sum approximation and symmetric approximation for spectral decomposition—it achieves runtime O(m · poly(log n, r, 1/ε)), offering a polynomial speedup over classical methods under low-rank constraints and enabling efficient entry-wise access and spectral decomposition of the solution matrix.

ABSTRACT

Semidefinite programming (SDP) is a central topic in mathematical optimization with extensive studies on its efficient solvers. In this paper, we present a proof-of-principle sublinear-time algorithm for solving SDPs with low-rank constraints; specifically, given an SDP with $m$ constraint matrices, each of dimension $n$ and rank $r$, our algorithm can compute any entry and efficient descriptions of the spectral decomposition of the solution matrix. The algorithm runs in time $O(m\cdot\mathrm{poly}(\log n,r,1/\varepsilon))$ given access to a sampling-based low-overhead data structure for the constraint matrices, where $\varepsilon$ is the precision of the solution. In addition, we apply our algorithm to a quantum state learning task as an application. Technically, our approach aligns with 1) SDP solvers based on the matrix multiplicative weight (MMW) framework by Arora and Kale [TOC '12]; 2) sampling-based dequantizing framework pioneered by Tang [STOC '19]. In order to compute the matrix exponential required in the MMW framework, we introduce two new techniques that may be of independent interest: $\bullet$ Weighted sampling: assuming sampling access to each individual constraint matrix $A_{1},\ldots,A_τ$, we propose a procedure that gives a good approximation of $A=A_{1}+\cdots+A_τ$. $\bullet$ Symmetric approximation: we propose a sampling procedure that gives the \emph{spectral decomposition} of a low-rank Hermitian matrix $A$. To the best of our knowledge, this is the first sampling-based algorithm for spectral decomposition, as previous works only give singular values and vectors.

Motivation & Objective

• To develop a classical algorithm that achieves sublinear runtime for low-rank semidefinite programs (SDPs), matching the exponential speedup previously seen in quantum algorithms.
• To address the challenge of computing matrix exponentials and spectral decompositions efficiently in the matrix multiplicative weight (MMW) framework using only sampling access to constraint matrices.
• To provide a succinct, efficient representation of the SDP solution that enables fast entry-wise queries and spectral information extraction.
• To demonstrate practical applicability by applying the algorithm to quantum state learning via shadow tomography.
• To establish a classical alternative to quantum SDP solvers under low-rank assumptions, leveraging dequantization techniques inspired by Tang's breakthrough.

Proposed method

• Leverages the matrix multiplicative weight (MMW) framework to iteratively update weight matrices and Gibbs states, approximating the optimal solution of the SDP.
• Introduces a novel weighted sampling procedure to approximate the sum of low-rank Hermitian matrices A1 + ... + Aτ using sampling access to individual matrices.
• Proposes a symmetric approximation technique that samples from the spectral decomposition of a low-rank Hermitian matrix, providing both eigenvalues and eigenvectors via sampling.
• Employs a data structure with sampling and norm query access to matrices, enabling O(poly(log n)) time per operation, crucial for sublinear complexity.
• Uses randomized estimation via Algorithms 4–6 to approximate Tr[Ajtρt] at each iteration, with high probability and bounded error.
• Applies binary search over the objective value using feasibility subroutines to solve the full optimization problem, while maintaining a succinct solution representation.

Experimental results

Research questions

• RQ1 Can a classical algorithm achieve sublinear runtime for low-rank SDPs under sampling access to constraint matrices, matching the exponential speedup of quantum algorithms?
• RQ2 Is it possible to sample from the spectral decomposition of a low-rank Hermitian matrix using only sampling access to its entries, rather than full matrix access?
• RQ3 Can the matrix exponential and Gibbs state computations required in the MMW framework be approximated efficiently using sampling-based techniques?
• RQ4 How can the solution to an SDP be represented succinctly to allow efficient entry-wise access and spectral decomposition?
• RQ5 Can this classical approach be applied to quantum state learning tasks such as shadow tomography with comparable efficiency to quantum algorithms?

Key findings

• The algorithm runs in time O(m · poly(log n, r, 1/ε)) for low-rank SDPs with m constraint matrices of dimension n and rank r, achieving sublinear dependence on n.
• It provides a succinct representation of the solution matrix, including its top eigenvalues, eigenvectors, and any specific entry, enabling efficient query access.
• The spectral decomposition is approximated via a novel symmetric sampling procedure, which is the first sampling-based method to recover both eigenvalues and eigenvectors of a low-rank Hermitian matrix.
• The algorithm achieves high-probability approximation of Tr[Ajtρt] using randomized estimation with error bounds controlled by ϵ and δ.
• The method is applied to shadow tomography, solving the problem in time O(m · poly(log n, 1/ϵ, log(1/δ), r)) with high probability.
• The runtime complexity is dominated by the spectral decomposition step, with a polynomial overhead in r and 1/ϵ, though this may be reducible via finer analysis.